HW1-sol - PHYSICS 4455 QUANTUM MECHANICS Problem Set 1...

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PHYSICS 4455 — QUANTUM MECHANICS Problem Set 1 — Solutions. 1. Recapping angular momentum. Liboff Problem 1.5 In cartesian coordinates, the z -component of angular momentum is given by L z = xp y yp x where p x = mx and p y = my . To express it in spherical coordinates, we use the transformation equations x = r cos φ sin θ , y = r sin φ sin θ , z = r cos θ and the corresponding derivatives: x = r cos φ sin θ− r φ sin φ sin θ+ r θ cos φ cos θ y = r sin φ sin r φ cos φ sin r θ sin φ cos θ ż = r cos r θ sin θ Putting it all together, we find L z = mxy myx = mr cos φ sin θ r sin φ sin r φ cos φ sin r θ sin φ cos θ mr sin φ sin θ r cos φ sin r φ sin φ sin r θ cos φ cos θ = mr 2 φ cos 2 φ sin 2 mr 2 φ sin 2 φ sin 2 θ = mr 2 φ sin 2 θ= p φ The last equality follows from Eqn. (1.19). Liboff Problem 1.7 The energy of a free particle is just kinetic energy: H = T = 1 2 mv 2 = p 2 /( 2 m ) .For simplicity, we start with the form given in Liboff and work our way backwards. H = p r 2 2 m + L 2 2 mr 2 Using p r =( p r )/ r and L 2 r × p ) 2 = r 2 p 2 −( r p ) 2 ,wefind H = ( p r ) 2 2 mr 2 + r 2 p 2 r p ) 2 2 mr 2 = r 2 p 2 2 mr 2 = p 2 2 m If you don’t like to work backwards, the procedure is to express p 2 = p x 2 + p y 2 + p z 2 in terms of p r , p θ ,and p φ (cf. Eq. (1.33)), and to do the same with L 2 - which is what we’ll do next. Note that the relation p 2 2 m = p r 2 2 m + L 2 2 mr 2 is not restricted to a free particle. We’ll use it again below.
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Liboff Problem 1.8. Here, we’ll express L 2 in terms of p r , p θ ,and p φ . You can short-cut this by using Eq.
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HW1-sol - PHYSICS 4455 QUANTUM MECHANICS Problem Set 1...

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